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{\Large \textbf{Sample Questions: Moment-generating functions}}

STA256 Fall 2019. Copyright information is at the end of the last page.
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\item {\Large Let $X$ have a moment-generating function $M_{_X}(t)$ and let $a$ be a constant. Show $M_{_{aX}}(t) = M_{_X}(at)$. }
    
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\item {\Large Let $X$ have a moment-generating function $M_{_X}(t)$ and let $a$ be a constant. Show $M_{_{a+X}}(t) = e^{at}M_{_X}(t)$. }

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\item {\Large Let $X$ and $Y$ be independent, (continuous) random variables.  Show $M_{_{X+Y}}(t) = M_{_X}(t) \, M_{_Y}(t)$. }


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\item {\Large Let $X \sim N(0,1)$. Calculate $M_{_X}(t)$. }

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\item {\Large Let $X \sim N(\mu,\sigma^2)$. Calculate $M_{_X}(t)$. }

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\item {\Large Let $X \sim N(\mu,\sigma^2)$. Show $Y = \frac{X-\mu}{\sigma} \sim N(0,1)$ using moment-generating functions.} 

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\item {\Large  Let $X \sim N(\mu,\sigma^2)$. Find the distribution of  $Y = a + bX$ using moment-generating functions.} 


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\item {\Large Let $X_1 \sim N(\mu_1,\sigma^2_1)$ and $X_2 \sim N(\mu_2,\sigma^2_2)$ be independent. Find the distribution of  $Y = aX_1+bX_2 + c$}.


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\item {\Large Let $Z \sim N(0,1)$ and let $Y = Z^2$. Find the distribution of  $Y$. Note that the MGF of a chi-squared random variable is $M(t)=(1-2t)^{-\frac{\nu}{2}}$.} 


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\item {\Large Independently for $i=1, \ldots, n$, let $Y_i \sim \chi^2(\nu_i)$. Find the distribution of $W=\sum_{i=1}^n Y_i$.} 


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\item {\Large Independently for $i=1, \ldots, n$, let $X_i \sim N(\mu_i,\sigma_i)$. What is the distribution of $Y=\sum_{i=1}^n \left(\frac{X_i-\mu_i}{\sigma_i}\right)^2$? Justify your answer.} 

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This handout was prepared by  \href{http://www.utstat.toronto.edu/~brunner}{Jerry Brunner},
Department of Mathematical and Computational Sciences, University of Toronto. It is licensed under a 
\href{http://creativecommons.org/licenses/by-sa/3.0/deed.en_US}
     {Creative Commons Attribution - ShareAlike 3.0 Unported License}. Use any part of it as you like and share the result freely. The \LaTeX~source code is available from the course website: 
     
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\href{http://www.utstat.toronto.edu/~brunner/oldclass/256f19} {\small\texttt{http://www.utstat.toronto.edu/$^\sim$brunner/oldclass/256f19}}
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